Steering quantum-memory-assisted entropic uncertainty under unital and nonunital noises via filtering operations

In this work, we investigate the dynamic features of the entropic uncertainty for two incompatible measurements under local unital and nonunital channels. Herein, we choose Pauli operators \(\sigma _x \) and \(\sigma _z \) as a pair of observables of interest measuring on particle A, and the uncertainty can be predicted when particle A is entangled with quantum memory B. We explore the dynamics of the uncertainty for the measurement under local unitary (phase-damping) and nonunitary (amplitude-damping) channels, respectively. Remarkably, we derive the entropic uncertainty relation under three different kinds of measurements of Pauli-observable pair under various realistic noisy environments; it has been found that the entropic uncertainty has the same tendency of its evolution during the AD and PD channel when we choose \(\sigma _x \) and \(\sigma _y \) measurement. Besides, we find out that the entropic uncertainty will have an optimal value if one chooses \(\sigma _x \) and \(\sigma _z \) as the measurement incompatibility, comparing with others. Furthermore, in order to reduce the entropic uncertainty in noisy environment, we propose an effective strategy to steer the amount by means of implementing a filtering operation on the particle under the two types of channels, respectively. It turns out that this operation can greatly reduce the entropic uncertainty by modulation of the operation strength. Thus, our investigations might offer an insight into the dynamics and steering of the entropic uncertainty in an open system.     

Entropic uncertainty relations in the Heisenberg <Emphasis Type="Italic">XXZ</Emphasis> model and its controlling via filtering operations

The uncertainty principle is recognized as an elementary ingredient of quantum theory and sets up a significant bound to predict outcome of measurement for a couple of incompatible observables. In this work, we develop dynamical features of quantum memory-assisted entropic uncertainty relations (QMA-EUR) in a two-qubit Heisenberg XXZ spin chain with an inhomogeneous magnetic field. We specifically derive the dynamical evolutions of the entropic uncertainty with respect to the measurement in the Heisenberg XXZ model when spin A is initially correlated with quantum memory B. It has been found that the larger coupling strength \( J \) of the ferromagnetism (\( J < 0 \)) and the anti-ferromagnetism (\( J > 0 \)) chains can effectively degrade the measuring uncertainty. Besides, it turns out that the higher temperature can induce the inflation of the uncertainty because the thermal entanglement becomes relatively weak in this scenario, and there exists a distinct dynamical behavior of the uncertainty when an inhomogeneous magnetic field emerges. With the growing magnetic field \( \left| B \right| \), the variation of the entropic uncertainty will be non-monotonic. Meanwhile, we compare several different optimized bounds existing with the initial bound proposed by Berta et al. and consequently conclude Adabi et al.’s result is optimal. Moreover, we also investigate the mixedness of the system of interest, dramatically associated with the uncertainty. Remarkably, we put forward a possible physical interpretation to explain the evolutionary phenomenon of the uncertainty. Finally, we take advantage of a local filtering operation to steer the magnitude of the uncertainty. Therefore, our explorations may shed light on the entropic uncertainty under the Heisenberg XXZ model and hence be of importance to quantum precision measurement over solid state-based quantum information processing.     

Wigner–Yanase skew information and entanglement generation in quantum measurement

The first step of quantum measurement procedure is known as premeasurement, during which correlation is established between the system and the measurement apparatus. Such correlation may be classical or nonclassical in nature. One compelling nonclassical correlation is entanglement, a useful resource for various quantum information theoretic protocols. Quantifying the amount of entanglement, generated during quantum measurement, therefore, seeks importance from practical ground, and this is the central issue of the present paper. Interestingly, for a two-level quantum system, we obtain that the amount of entanglement, measured in term of negativity, generated in premeasurement process can be quantified by two factors: skew information, which quantifies the uncertainty in the measurement of an observable not commuting with some conserved quantity of the system, and mixedness parameter of the system’s initial state.     

Quantum-memory-assisted entropic uncertainty in two-qubit Heisenberg <Emphasis Type="Italic">XYZ</Emphasis> chain with Dzyaloshinskii–Moriya interactions and effects of intrinsic decoherence

Quantum-memory-assisted entropic uncertainty relation (QMA-EUR) in two-qubit Heisenberg XYZ spin chain model with Dzyaloshinskii–Moriya (DM) interaction has been investigated. The paper shows that the DM interactions and the spin interactions alone x, y, z directions can efficiently suppress the entropic uncertainty of Pauli observables (\(\sigma _{x}\) and \(\sigma _{z}\)), even make the entropic uncertainty close to zero. As well, it is pointed out that the entropic uncertainty reaches to zero at very low temperature, starts to increase with temperature after a threshold, and generally becomes constant at a fixed value. We also verified the Bob’s uncertainty about Alice’s measurement outcomes is anticorrelated with the sum of the accessible information of observer. Furthermore, the decoherence conditions including dephasing and noisy environments are considered. For the fixed initial state, the entropic uncertainty of the XYZ model with DM interaction in z-direction are independent of spin–spin coupling \(J_z\) and the anisotropy parameter \(\varDelta \). In the dephasing environment, the evolutions of entropic uncertainty and its lower bound \(U_{B}\) oscillate with the time and saturates at a finite value, and this value is varied with the purity parameter r of initial state. In the noisy environment, the entropic uncertainty and its lower bound monotonically increase with the time and will be stable at value 2 quickly. This is because the combined effects of the DM interaction and the decoherence force the various initial entanglement states to oscillate into an identical state, regardless of the value of \(D_{z}\) and the parameter r of initial state.     

Exploring entropic uncertainty relation in the Heisenberg <Emphasis Type="Italic">XX</Emphasis> model with inhomogeneous magnetic field

In this work, we investigate the quantum-memory-assisted entropic uncertainty relation in a two-qubit Heisenberg XX model with inhomogeneous magnetic field. It has been found that larger coupling strength J between the two spin-chain qubits can effectively reduce the entropic uncertainty. Besides, we observe the mechanics of how the inhomogeneous field influences the uncertainty, and find out that when the inhomogeneous field parameter \(b<1\), the uncertainty will decrease with the decrease of the inhomogeneous field parameter b, conversely, the uncertainty will increase with decreasing b under the condition that \(b>1\). Intriguingly, the entropic uncertainty can shrink to zero when the coupling coefficients are relatively large, while the entropic uncertainty only reduces to 1 with the increase of the homogeneous magnetic field. Additionally, we observe the purity of the state and Bell non-locality and obtain that the entropic uncertainty is anticorrelated with both the purity and Bell non-locality of the evolution state.     

Entropic uncertainty relation in a two-qutrit system with external magnetic field and Dzyaloshinskii–Moriya interaction under intrinsic decoherence

In this paper, we explore the dynamic behaviors of entropic uncertainty relation in a two-qutrit system which is in the presence of external magnetic field and Dzyaloshinskii–Moriya (DM) interaction under intrinsic decoherence. The effects of the isotropic bilinear interaction, the external magnetic field, the DM interaction strength, as well as the intrinsic decoherence on the entropic uncertainty relation have been demonstrated in detail. Compared with previous results, our results show that, controlling the isotropic bilinear interaction parameter J, the external magnetic field strength \(B_{0}\), the DM interaction parameter D can result in inflation of the uncertainty, while increasing the intrinsic decoherence parameter can lift the uncertainty of the measurement. In particularly, under certain conditions (e.g., parameters J, \(B_{0}\) and D are large enough), the entropic uncertainty will ultimately tend to a stable value and be immune to decoherence.     

Security of a semi-quantum protocol where reflections contribute to the secret key

In this paper, we provide a proof of unconditional security for a semi-quantum key distribution protocol introduced in a previous work. This particular protocol demonstrated the possibility of using X basis states to contribute to the raw key of the two users (as opposed to using only direct measurement results) even though a semi-quantum participant cannot directly manipulate such states. In this work, we provide a complete proof of security by deriving a lower bound of the protocol’s key rate in the asymptotic scenario. Using this bound, we are able to find an error threshold value such that for all error rates less than this threshold, it is guaranteed that A and B may distill a secure secret key; for error rates larger than this threshold, A and B should abort. We demonstrate that this error threshold compares favorably to several fully quantum protocols. We also comment on some interesting observations about the behavior of this protocol under certain noise scenarios.     

An uncertainty relation in terms of generalized metric adjusted skew information and correlation measure

The uncertainty principle in quantum mechanics is a fundamental relation with different forms, including Heisenberg’s uncertainty relation and Schrödinger’s uncertainty relation. In this paper, we prove a Schrödinger-type uncertainty relation in terms of generalized metric adjusted skew information and correlation measure by using operator monotone functions, which reads,

$$\begin{aligned} U_\rho ^{(g,f)}(A)U_\rho ^{(g,f)}(B)\ge \frac{f(0)^2l}{k}\left| \mathrm {Corr}_\rho ^{s(g,f)}(A,B)\right| ^2 \end{aligned}$$

for some operator monotone functions f and g, all n-dimensional observables A, B and a non-singular density matrix \(\rho \). As applications, we derive some new uncertainty relations for Wigner–Yanase skew information and Wigner–Yanase–Dyson skew information.

     

Asymptotic confidence interval for conditional probability at decision making

Consideration was given to construction of a unilateral asymptotic confidence interval for an unknown conditional probability of the event A under the condition B. Analytical expressions for the boundaries of such interval were obtained for various a priori restrictions on the probabilities of the events A and B such that the probability of the event B is separable from zero by a certain value, as well that there exists an upper a priori restriction on the probability of the event A. The interval estimates obtained were compared in precision.     

Two-agent flowshop scheduling to maximize the weighted number of just-in-time jobs

We consider the scheduling problem in which two agents (agents A and B), each having its own job set (containing the A-jobs and B-jobs, respectively), compete to process their own jobs in a two-machine flowshop. Each agent wants to maximize a certain criterion depending on the completion times of its jobs only. Specifically, agent A desires to maximize either the weighted number of just-in-time (JIT) A-jobs that are completed exactly on their due dates or the maximum weight of the JIT A-jobs, while agent B wishes to maximize the weighted number of JIT B-jobs. Evidently four optimization problems can be formulated by treating the two agents’ criteria as objectives and constraints of the corresponding optimization problems. We focus on the problem of finding the Pareto-optimal schedules and present a bicriterion analysis of the problem. Solving this problem also solves the other three problems of bicriterion scheduling as a by-product. We show that the problems under consideration are either polynomially or pseudo-polynomially solvable. In addition, for each pseudo-polynomial-time solution algorithm, we show how to convert it into a two-dimensional fully polynomial-time approximation scheme for determining an approximate Pareto-optimal schedule. Finally, we conduct extensive numerical studies to evaluate the performance of the proposed algorithms.     

On Fixed Cost k-Flow Problems

In the Fixed Cost k-Flow problem, we are given a graph G = (V, E) with edge-capacities {u _e ∣e ∈ E} and edge-costs {c _e ∣e ∈ E}, source-sink pair s, t ∈ V, and an integer k. The goal is to find a minimum cost subgraph H of G such that the minimum capacity of an st-cut in H is at least k. By an approximation-preserving reduction from Group Steiner Tree problem to Fixed Cost k-Flow, we obtain the first polylogarithmic lower bound for the problem; this also implies the first non-constant lower bounds for the Capacitated Steiner Network and Capacitated Multicommodity Flow problems. We then consider two special cases of Fixed Cost k-Flow. In the Bipartite Fixed-Cost k-Flow problem, we are given a bipartite graph G = (A ∪ B, E) and an integer k > 0. The goal is to find a node subset S ? A ∪ B of minimum size |S| such G has k pairwise edge-disjoint paths between S ∩ A and S ∩ B. We give an \(O(\sqrt {k\log k})\) approximation for this problem. We also show that we can compute a solution of optimum size with Ω(k/polylog(n)) paths, where n = |A| + |B|. In the Generalized-P2P problem we are given an undirected graph G = (V, E) with edge-costs and integer charges {b _v : v ∈ V}. The goal is to find a minimum-cost spanning subgraph H of G such that every connected component of H has non-negative charge. This problem originated in a practical project for shift design [11]. Besides that, it generalizes many problems such as Steiner Forest, k-Steiner Tree, and Point to Point Connection. We give a logarithmic approximation algorithm for this problem. Finally, we consider a related problem called Connected Rent or Buy Multicommodity Flow and give a log^3+?? n approximation scheme for it using Group Steiner Tree techniques.     

Towards more realistic (e.g., non-associative) “and”- and “or”-operations in fuzzy logic

How is fuzzy logic usually formalized? There are many seemingly reasonable requirements that a logic should satisfy: e.g., since A B and B A are the same, the corresponding and-operation should be commutative. Similarly, since A A means the same as A, we should expect that the and-operation should also satisfy this property, etc. It turns out to be impossible to satisfy all these seemingly natural requirements, so usually, some requirements are picked as absolutely true (like commutativity or associativity), and others are ignored if they contradict to the picked ones. This idea leads to a neat mathematical theory, but the analysis of real-life expert reasoning shows that all the requirements are only approximately satisfied. we should require all of these requirements to be satisfied to some extent. In this paper, we show the preliminary results of analyzing such operations. In particular, we show that non-associative operations explain the empirical 7±2 law in psychology according to which a person can normally distinguish between no more than 7 plus minus 2 classes.     

Negativity in the generalized Valence Bond Solid state

Using a graphical presentation of the spin S one-dimensional Valence Bond Solid (VBS) state, based on the representation theory of the \({\textit{SU}}(2)\) Lie algebra of spins, we compute the spectrum of a mixed-state reduced density matrix. This mixed state of two blocks of spins A and B is obtained by tracing out the spins outside A and B, in the pure VBS state density matrix. We find in particular that the negativity of the mixed state is nonzero only for adjacent subsystems. The method introduced here can be generalized to the computation of entanglement properties in Levin–Wen models, that possess a similar algebraic structure to the VBS state in the ground state.     

Solvability conditions of problem of synthesizing proportional-integral control of a linear MIMO system

The solvability conditions and just the solution of the problem of the regular and irregular proportional-integral (PI) control are found in accordance with the properties of invariant zeros of a multi-input multioutput (MIMO) system. It is proved that the problem of synthesizing the control of the MIMO system is solvable if and only if the pair of matrices (A, B) that describes a control plant is controllable and the matrix B_L^⊥AC_R^⊥ (where B_L^⊥ is the left zero divisor of the matrix B and C_R^⊥ is the right zero divisor of the output matrix C) has a complete row rank.     

Position-based coding and convex splitting for private communication over quantum channels

The classical-input quantum-output (cq) wiretap channel is a communication model involving a classical sender X, a legitimate quantum receiver B, and a quantum eavesdropper E. The goal of a private communication protocol that uses such a channel is for the sender X to transmit a message in such a way that the legitimate receiver B can decode it reliably, while the eavesdropper E learns essentially nothing about which message was transmitted. The \(\varepsilon \)-one-shot private capacity of a cq wiretap channel is equal to the maximum number of bits that can be transmitted over the channel, such that the privacy error is no larger than \(\varepsilon \in (0,1)\). The present paper provides a lower bound on the \(\varepsilon \)-one-shot private classical capacity, by exploiting the recently developed techniques of Anshu, Devabathini, Jain, and Warsi, called position-based coding and convex splitting. The lower bound is equal to a difference of the hypothesis testing mutual information between X and B and the “alternate” smooth max-information between X and E. The one-shot lower bound then leads to a non-trivial lower bound on the second-order coding rate for private classical communication over a memoryless cq wiretap channel.     

Partitions of the set of selected unknowns in linear differential–algebraic systems

As was shown earlier, for a linear differential–algebraic system A ₁ y′ + A ₀ y = 0 with a selected part of unknowns (entries of a column vector y), it is possible to construct a differential system ?′ = B ?, where the column vector ? is formed by some entries of y, and a linear algebraic system by means of which the selected entries that are not contained in ? can be expressed in terms of the selected entries included in ?. In the paper, sizes of the differential and algebraic systems obtained are studied. Conditions are established under the fulfillment of which the size of the algebraic system is determined unambiguously and the size of the differential system is minimal.     

On Interval Systems [<Emphasis Type="Italic">x</Emphasis>] = [<Emphasis Type="Italic">A</Emphasis>][<Emphasis Type="Italic">x</Emphasis>] + [<Emphasis Type="Italic">b</Emphasis>] and the Powers of Interval Matrices in Complex Interval Arithmetics

For the interval system of equations defined by [x] = [A][x] + [b] we derive necessary and sufficient criteria for the existence of solutions [x]. Furthermore we give necessary and sufficient criteria for the convergence of powers of [A]. In contrast to former results we treat complex interval arithmetics.     

The <Emphasis Type="Italic">B</Emphasis><Superscript><Emphasis Type="Italic">dual</Emphasis></Superscript>-Tree: indexing moving objects by space filling curves in the dual space

Existing spatiotemporal indexes suffer from either large update cost or poor query performance, except for the B ^x -tree (the state-of-the-art), which consists of multiple B ⁺-trees indexing the 1D values transformed from the (multi-dimensional) moving objects based on a space filling curve (Hilbert, in particular). This curve, however, does not consider object velocities, and as a result, query processing with a B ^x -tree retrieves a large number of false hits, which seriously compromises its efficiency. It is natural to wonder “can we obtain better performance by capturing also the velocity information, using a Hilbert curve of a higher dimensionality?”. This paper provides a positive answer by developing the B ^dual -tree, a novel spatiotemporal access method leveraging pure relational methodology. We show, with theoretical evidence, that the B ^dual -tree indeed outperforms the B ^x -tree in most circum- stances. Furthermore, our technique can effectively answer progressive spatiotemporal queries, which are poorly supported by B ^x -trees.     

Restricted (<Emphasis Type="Italic">k</Emphasis>, <Emphasis Type="Italic">n</Emphasis>)-threshold quantum secret sharing scheme based on local distinguishability of orthogonal multiqudit entangled states

In this work, we study a restricted (k, n)-threshold access structure. According to this structure, we construct a group of orthogonal multipartite entangled states in d-dimensional system and investigate the distinguishability of these entangled states under restricted local operations and classical communication. Based on these properties, we propose a restricted (k, n)-threshold quantum secret sharing scheme (called LOCC-QSS scheme). The k cooperating players in the restricted threshold scheme come from all disjoint groups. In the proposed protocol, the participants distinguish these orthogonal states by the computational basis measurement and classical communication to reconstruct the original secret. Furthermore, we also analyze the security of our scheme in three primary quantum attacks and give a simple encoding method in order to better prevent the participant conspiracy attack.     

Selective A<Emphasis Type="Italic">n</Emphasis>DE for large data learning: a low-bias memory constrained approach

Learning from data that are too big to fit into memory poses great challenges to currently available learning approaches. Averaged n-Dependence Estimators (AnDE) allows for a flexible learning from out-of-core data, by varying the value of n (number of super parents). Hence, AnDE is especially appropriate for learning from large quantities of data. Memory requirement in AnDE, however, increases combinatorially with the number of attributes and the parameter n. In large data learning, number of attributes is often large and we also expect high n to achieve low-bias classification. In order to achieve the lower bias of AnDE with higher n but with less memory requirement, we propose a memory constrained selective AnDE algorithm, in which two passes of learning through training examples are involved. The first pass performs attribute selection on super parents according to available memory, whereas the second one learns an AnDE model with parents only on the selected attributes. Extensive experiments show that the new selective AnDE has considerably lower bias and prediction error relative to A\(n'\)DE, where \(n' = n-1\), while maintaining the same space complexity and similar time complexity. The proposed algorithm works well on categorical data. Numerical data sets need to be discretized first.